To determine the missing exponent of [tex]\( y \)[/tex] in the second term for the polynomial [tex]\( 6xy^2 - 5x^2y^? + 9x^2 \)[/tex] to be a trinomial with a degree of 3, let's analyze each term in detail:
1. First Term: [tex]\( 6xy^2 \)[/tex]
- The term [tex]\( 6xy^2 \)[/tex] can be rewritten as [tex]\( 6x^1y^2 \)[/tex].
- The degree of this term is the sum of the exponents of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
\text{Degree} = 1 + 2 = 3
\][/tex]
2. Second Term: [tex]\( -5x^2y^? \)[/tex]
- Let's denote the missing exponent of [tex]\( y \)[/tex] as [tex]\( b \)[/tex]. Therefore, the term becomes [tex]\( -5x^2y^b \)[/tex].
- To find the correct value of [tex]\( b \)[/tex] such that this term has a degree of 3, we note:
[tex]\[
\text{Degree} = 2 + b = 3
\][/tex]
- Solving for [tex]\( b \)[/tex]:
[tex]\[
b = 3 - 2 = 1
\][/tex]
- Hence, the exponent of [tex]\( y \)[/tex] should be [tex]\( 1 \)[/tex], making the term [tex]\( -5x^2y \)[/tex].
3. Third Term: [tex]\( 9x^2 \)[/tex]
- The term [tex]\( 9x^2 \)[/tex] can be rewritten as [tex]\( 9x^2y^0 \)[/tex].
- The degree of this term is the sum of the exponents of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
\text{Degree} = 2 + 0 = 2
\][/tex]
To ensure this polynomial [tex]\( 6xy^2 - 5x^2y + 9x^2 \)[/tex] is considered a trinomial with a degree of 3, we check the degree of each term:
- The first term [tex]\( 6xy^2 \)[/tex] has a degree of 3.
- The second term [tex]\( -5x^2y \)[/tex] has a degree of 3.
- The third term [tex]\( 9x^2 \)[/tex] has a degree of 2, which is less than 3 and thus fitting the requirement.
This confirms that the polynomial [tex]\( 6xy^2 - 5x^2y + 9x^2 \)[/tex] is a trinomial with a maximum degree of 3 once the missing exponent [tex]\( b \)[/tex] is set to [tex]\( 1 \)[/tex].
Therefore, the missing exponent of [tex]\( y \)[/tex] in the second term is [tex]\( \boxed{1} \)[/tex].
